kinetics of a particlewe have seen the effect of

Question

Kinetics of a Particle:

We have seen the effect of forces on a particle. The laws of motion for a particle are extended to include a system of particles or a rigid body. It is done by using D′Alembert's principle. D′Almbert's principle states that the resultant of forces acting on a body is equivalent to the sum of the effective forces acting on all of the particles of the body.

Further, we may say that while forces are acting on the particles of a body, we may imagine inertia forces also acting on the particles bringing the body in dynamic equilibrium. This principle may be extended further to find out the motion of mass centre of any body. We understand that the resultant of the applied external forces is equal to the product of the mass of the body and the acceleration of its mass centre which is expressed as F = ma.

The motion of translation of a rigid body is described as a motion in which a straight line passing through any two particles of a body is always parallel to its initial position. Further, we may say that the translation of a rigid body is equivalent to that of a particle having the mass of the body and the motion of the mass centre of the body.

For rectilinear motion, Consider x axis be directed along the initial direction of motion, and reckoned as positive. Then we can write the equations as below,

∑ F_x  = ma.

∑ F_y  = 0,       ∑ F_z  = 0

 For curvilinear motion, normal and tangential components of acceleration (N and T, respectively) are utilized to write equations.

∑ F _n = m a_n  = m (v²/r)

∑ F_t = m a_t = m( dv/ dt)

The problem becomes very simple while inertia reactions of magnitudes m v ² /r and m . dv/ dt are applied via the mass centre and directed opposite to the normal and tangential components of acceleration respectively and dynamic equilibrium equations are written.

The advantage of dynamic equilibrium is that all the methods of statics may be applied to a single free body diagram which includes the components of inertial reaction ma. This is very much useful while moment summation is calculated at a centre which is the intersection of two unknown forces and set to zero.